Fractional Analogue of Sturm–Liouville Operator

Niyaz Tokmagambetov, Berikbol T. Torebek

Received: March 3, 2016 Revised: August 30, 2016

Communicated by Christian Bär

Abstract. In this paper we study a symmetric fractional differential operator of order 2α, (1/2 < α < 1). Using the extension theory a class of self–adjoint problems generated by the fractional Sturm– Liouville equation is described.

2010 Mathematics Subject Classification: Primary 26A33; Secondary 34L10 Keywords and Phrases: Self–adjoint operator, symmetric operator, fractional Sturm–Liouville operator, fractional differential equation, boundary value problem, boundary condition, Caputo operator, Riemann–Liouville operator.

1. Introduction

Many physical processes (diffusive processes, thermal processes and etc.) are expressed by fractional differential equations. Meanwhile, the study of boundary value problems for differential equations of fractional order is also very important to enrich and improve the fractional calculus theory. The fractional calculus has been an active field of research during several decades. In particular, the Mittag-Leffler functions are well-known in the theory of the fractional calculus, which allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The basic properties are proved in [DN59]. Further investigations were done by Kilbas and Trujillo [KT02], Popov [P02], Jin and Rundell [JR12], and others. For more details we refer to [PS11, GKMR14] and references therein. However, there are more open questions, for example in the spectral theory. It is well-known that the classical Sturm–Liouville equation

(1.1)  Su(x) ≡ u′′(x) + q(x)u(x), x ∈ (a, b)

with real q ∈ C^{1}[a, b] and with boundary conditions

u(a) = 0, u(b) = 0

is a self–adjoint operator in L^{2}(a, b). Indeed, there is a class of so called ‘strongly regular’ boundary conditions [N67] which produce self–adjoint operators. Nevertheless, it is unclear how to formulate a fractional analogue. Roughly speaking, fractional differential equations with the classical boundary conditions are not self–adjoint in the Hilbert space. Since self–adjointness implies a basis property of the system of root functions, mathematicians also were interested in approximation properties of fractional differential operators. For instance, the system of root functions for a fractional Sturm–Liouville type operator was investigated in [D70]. In [N77, A82] the authors studied spectral properties of the Sturm–Liouville equation with lower order fractional derivatives. More recent results can be found in [M10, RTV13, DWF14, P14, A15]. However, only non self–adjoint problems were considered in all of these papers. Fortunately, Klimek and Agrawal [KA13] found a symmetric fractional operator in the special weighted space of continuous functions. However, finding of new symmetric fractional operators is still interesting. In this work we aim to find a symmetric fractional operator in the Hilbert space. Given a fractional differential equation of order 2α, (1/2 < α < 1), on an interval (a, b), the main issue is to choose ‘suitable’ boundary conditions to get a symmetric operator. Here, we define boundary functionals and obtain a symmetric fractional Sturm–Liouville operator in a ‘suitable’ Hilbert space. Using the extension theory of operators a class of self–adjoint problems is described. Finally, we derive spectral properties and allocate positive operators from the self–adjoint operators. For applications of symmetric fractional operators to the related topics, see [KOM14, BC14, KDE15, LQ15, KM16], and for numerical realizations we refer to [AS10, ZK13, HMA14]. In subsequent works we will apply Fourier Analysis technics (see, for instance [ZK14, K15, QDH15]) in a combination with the self–adjoint fractional Sturm– Liouville operators obtained here to solve mixed problems of sub–diffusion, super–diffusion, anomalous diffusion, fractional Laplacian and others.

2. Main Results

In this section, we state main results and compile some basic definitions of fractional differential operators. For a fuller treatment the reader is referred to [SKM87, KST06] and references therein. Now we give definitions of the Riemann–Liouville fractional integrals and derivatives, and formulate the Caputo fractional derivatives. Also, we will use the sequential differentiation introduced in ([KST06], p. 394).

Definition 2.1. The left and right Riemann–Liouville fractional integrals I_{a+}^{α} and I_{b−}^{α} of order α ∈ ℝ (α > 0) are given by

I_{a+}^{α} [f] (t) = \frac{1}{Γ (α)} ∫_{a}^{t} (t − s)^{α−1} f (s)ds, t ∈ (a, b],

Fractional Analogue of Sturm–Liouville Operator 1505

and

I_{b−}^{α} [f] (t) = \frac{1}{Γ (α)} ∫_{t}^{b} (s − t)^{α−1} f (s)ds, t ∈ [a, b),

respectively. Here Γ denotes the Euler gamma function.

Definition 2.2. The left Riemann–Liouville fractional derivative D_{a+}^{α} of order α ∈ ℝ (0 < α < 1) is defined by

D_{a+}^{α} [f] (t) = \frac{d}{dt}I_{a+}^{1−α} [f] (t) , ∀t ∈ (a, b].

Similarly, the right Riemann–Liouville fractional derivative D_{b−}^{α} of order α ∈ ℝ (0 < α < 1) is given by

D_{b−}^{α} [f] (t) = −\frac{d}{dt}I_{b−}^{1−α} [f] (t) , ∀t ∈ [a, b).

Definition 2.3. The left and right Caputo fractional derivatives of order α ∈ ℝ (0 < α < 1) are defined by

D_{a+}^{α} [f] (t) = D_{a+}^{α} [f (t) − f (a)] , t ∈ (a, b],

and

D_{b−}^{α} [f] (t) = D_{b−}^{α} [f (t) − f (b)] , t ∈ [a, b),

respectively.

Consider the expression

(2.1)  Lu := D_{a+}^{α} (D_{b−}^{α} (u))

in L^{2}(a, b). Here we assume that \frac{1}{2} < α < 1. Now, we define and characterize a space generated by the Caputo–Riemann– Liouville equation (2.1).

Theorem 2.1. The space

H^{2α}(a, b) = {u ∈ L^{2α}(a, b) : D_{a+}^{α}D_{b−}^{α}u ∈ L^{2}(a, b)}

closed with respect to the norm

∥u∥_{H^{2α}(a,b)} := ∥u∥_{L^{2}(a,b)} + ∥D_{a+}^{α}D_{b−}^{α}u∥_{L^{2}(a,b)}

is a Banach space. Here, L^{2α}(a, b) is a Hölder space of the order 2α. Furthermore, H^{2α}(a, b) is the Hilbert space with the inner product

(u, v)_{H^{2α}(a,b)} := (u, v) + (D_{a+}^{α}D_{b−}^{α}u, D_{a+}^{α}D_{b−}^{α}v),

where (·, ·) is the inner product of the space L^{2}(a, b).

Define L_{m} as an operator acting from L^{2}(a, b) to L^{2}(a, b) by the formula (2.1) with the domain

D(L_{m}) = {u ∈ H^{2α}(a, b) : ξ_{1}^{−}(u) = ξ_{2}^{−}(u) = ξ_{1}^{+}(u) = ξ_{2}^{+}(u) = 0} ,

where functionals ξ_{1}^{−}(u), ξ_{2}^{−}(u), ξ_{1}^{+}(u), ξ_{2}^{+}(u) are defined as follows:

(2.2)  \begin{matrix}ξ_{1}^{−}(u) := I_{b−}^{1−α} [u] (a) , ξ_{2}^{−}(u) := I_{b−}^{1−α} [u] (b) , \\ ξ_{1}^{+}(u) := D_{b−}^{α} [u] (a) , ξ_{2}^{+}(u) := D_{b−}^{α} [u] (b) . \end{matrix}

Also, we introduce the operator by the action (2.1)

L_{M} : L^{2}(a, b) ⟶ L^{2}(a, b)

with the domain D(L_{M}) := {u ∈ H^{2α}(a, b)}. Note that to investigate the time–fractional diffusion equation the authors of [GLY15] used the fractional Sobolev space explored by Adams [A99]. They showed that the space is equivalent to the Hilbert space induced by a second order differential operator. Using the following class of matrices, we describe self-adjoint problems in Theorem 2.2. Also, an analogue of the strongly regular boundary conditions is obtained.

Definition 2.4. We say that

θ := ( \begin{matrix}θ_{11} θ_{12} θ_{13} θ_{14} \\ θ_{21} θ_{22} θ_{23} θ_{24} \end{matrix} )

is a SA–matrix if it can be represented in one of the following forms:

( \begin{matrix}1 0 r c \\ 0 1 − c d \end{matrix} ) , ( \begin{matrix}d 1 0 r \\ c 0 1 d \end{matrix} ) ,

( \begin{matrix}1 d r 0 \\ 0 c − d 1 \end{matrix} ) , ( \begin{matrix}r c 1 0 \\ −c d 0 1 \end{matrix} ) ,

for r, c, d ∈ ℝ. The matrices

( \begin{matrix}θ_{11} θ_{12} θ_{13} θ_{14} \\ θ_{21} θ_{22} θ_{23} θ_{24} \end{matrix} ) and ( \begin{matrix}θ_{21} θ_{22} θ_{23} θ_{24} \\ θ_{11} θ_{12} θ_{13} θ_{14} \end{matrix} )

are not distinguished.

Theorem 2.2. Let θ is a SA–matrix. Then L_{θ} introduced by

D_{a+}^{α}D_{b−}^{α}u(x) = f(x), a < x < b,

for u ∈ H^{2α}(a, b) with conditions

θ_{11}ξ_{1}^{−}(u) + θ_{12}ξ_{2}^{−}(u) + θ_{13}ξ_{1}^{+}(u) + θ_{14}ξ_{2}^{+}(u) = 0,
θ_{21}ξ_{1}^{−}(u) + θ_{22}ξ_{2}^{−}(u) + θ_{23}ξ_{1}^{+}(u) + θ_{24}ξ_{2}^{+}(u) = 0,

is a self–adjoint extension of L_{m} in H^{2α}(a, b).

Fractional Analogue of Sturm–Liouville Operator 1507

3. Properties of Fractional Operators

Now we formulate some well–known properties of fractional operators [SKM87, KST06]. Property 3.1. (cf. [KST06], p. 73, p. 76, p. 96) Let f ∈ L^{1}(a, b) and 0 < α, β < 1. Then, equations

I_{a+}^{α}I_{a+}^{β}f(x) = I_{a+}^{α+β}f(x),

I_{b−}^{α}I_{b−}^{β}f(x) = I_{b−}^{α+β}f(x),

and

I_{b−}^{α}D_{b−}^{α}f(x) = f(x) − I_{b−}^{1−α}f(a)\frac{(b − x)^{α−1}}{Γ(α)}

are satisfied a.e. in [a, b]. If a function f is absolutely continuous, then

I_{a+}^{α}D_{a+}^{α}f(x) = f(x) − f(a)

holds for almost all x ∈ [a, b]. Property 3.2. (cf. [SKM87], p. 87) Let α, β > 0, and C is a constant. Then for all ε ∈ [0, 1] the function

f(x) = C \frac{Γ(α + β)}{Γ(β)}(b − x − ε)_{∗}^{β−1} = \{ \begin{matrix}0, b − x ≤ ε; \\ C \frac{Γ(α+β)}{Γ(β)}(b − x − ε)^{β−1}, b − x > ε; \end{matrix}

satisfies

I_{b−}^{α}f(x) = \{ \begin{matrix}0, b − x ≤ ε; \\ C(b − x − ε)^{α+β−1} b − x > ε; \end{matrix}

almost everywhere on [a, b]. Property 3.3. Let 0 < α < 1 and f ∈ L^{2}(a, b). Then for arbitrary ε ∈ (a, b)

f(x) = C(b − x − ε)_{∗}^{α−1} = \{ \begin{matrix}0, b − x ≤ ε; \\ C(b − x − ε)^{α−1}, b − x > ε; \end{matrix}

satisfies

D_{b−}^{α}f(x) = 0

almost everywhere on [a, b]. Property 3.4. Let 0 < α < 1 and f ∈ L^{2}(a, b). Then for all ε ∈ (a, b) the function

f(x) = Cθ(b − x − ε) = \{ \begin{matrix}0, b − x ≤ ε, \\ C, b − x > ε, \end{matrix} C = const, a < x < b,

satisfies the equality

D_{a+}^{α}f(x) = 0, a < x < b,

where θ(x) is the Heaviside function.

Documenta Mathematica 21 (2016) 1503–1514

Property 3.5. Fix ε ∈ (a, b). Let 0 < α < 1 and f ∈ L^{2}(a, b). Then for any C_{1}, C_{2} the function

f(x) = C_{1}(b − x − ε)_{∗}^{α−1} + C_{2}(b − x − ε)_{∗}^{α},

satisfies

D_{b−}^{α}f(x) = C_{2}θ(b − x − ε)

for almost all x ∈ [a, b].

Property 3.6. (cf. [KST06], p. 76) Let u, v ∈ L^{2}(a, b) and 0 < α < 1. Then we have the fractional integration by parts formula

(I_{b−}^{β}u(t), v(t)) = (u(t), I_{a+}^{β}v(t)) .

4. Proofs

We begin by proving some necessary properties of the operators L_{m} and L_{M}.

Lemma 4.1. Fix ε ∈ [a, b]. A linear combination of (b−x−ε)_{∗}^{α} and (b−x−ε)_{∗}^{α−1} is from the kernel of L_{M} (KerL_{M}).

The proof of Lemma 4.1 follows from Properties 3.2, 3.3, 3.4 and 3.5.

Lemma 4.2. The equation L_{m}u = g has a solution u ∈ D(L_{m}) if and only if there exists f ∈ L^{2}(a, b) such that (f, v) = 0 for any v ∈ KerL_{M}:

R(L_{m}) ⊕ KerL_{M} = L^{2}(a, b).

Proof. Let f ∈ R(L_{m}). Then there exists w ∈ L^{2}(a, b) such that for any v ∈ KerL_{M} we have

(f, v) = (L_{m}w, v) = (w, L_{M}v) = 0.

Fix f ∈ L^{2}(a, b) with (f, v) = 0 for all v from KerL_{M}. By definition of L_{M} there is g ∈ Dom(L_{M}) such that L_{M}g = f. Easy to see that for an arbitrary function v ∈ KerL_{M} we obtain

(4.1)  0 = (f, v) = (L_{M}g, v) = ∑_{i=1}^{2}[ξ_{i}^{−}(v)ξ_{i}^{+}(g) − ξ_{i}^{−}(g)ξ_{i}^{+}(v)].

Indeed, informal calculations of (D_{a+}^{α} [D_{b−}^{α}u] , v) proves the last equality. By changing integration order in

(4.2) (D_{a+}^{α} [D_{b−}^{α}u] , v) = \frac{1}{Γ(1 − α)} ∫_{a}^{b} ∫_{a}^{x}(x − t)^{−α} \frac{d}{dt}D_{b−}^{α}u(t)dtv(x)dx,

we get

∫_{a}^{b} ∫_{a}^{x}(x − t)^{−α} \frac{d}{dt}D_{b−}^{α}u(t)dtv(x)dx

= ∫_{a}^{b} \frac{d}{dt}D_{b−}^{α}u(t) ∫_{t}^{b}(x − t)^{−α}v(x)dxdt. (4.3)

Integrating by parts in the right–side of the equation (4.3), we obtain

\frac{1}{Γ(1 − α)} ∫_{a}^{b} \frac{d}{dt}D_{b−}^{α}u(t) ∫_{t}^{b}(x − t)^{−α}v(x)dxdt =
= D_{b−}^{α}u(t)I_{b−}^{1−α}v(t)|_{a}^{b}+ (D_{b−}^{α}u, D_{b−}^{α}v) .

Let us calculate

(D_{b−}^{α}u, D_{b−}^{α}v) = − ∫_{a}^{b} \frac{d}{dt}I_{b−}^{1−α}u(t)D_{b−}^{α}v(t)dt

= −I_{b−}^{1−α}u(t)D_{b−}^{α}v(t)|_{a}^{b} + ∫_{a}^{b} I_{b−}^{1−α}u(t)\frac{d}{dx}D_{b−}^{α}v(t)dt.

Now, by applying the fractional integration by parts formula (Property 3.6)

(I_{b−}^{β}u, v) = (u, I_{a+}^{β}v)

to (I_{b−}^{1−α}u, \frac{d}{dx}D_{b−}^{α}v), we have

(I_{b−}^{1−α}u, \frac{d}{dx}D_{b−}^{α}v) = (u, D_{a+}^{α} [D_{b−}^{α}] v) .

As a result, we get

(D_{a+}^{α} [D_{b−}^{α}u] , v) − (u, D_{a+}^{α} [D_{b−}^{α}] v)
(4.4)  = D_{b−}^{α}u(t)I_{b−}^{1−α}v(t)|_{a}^{b} − I_{b−}^{1−α}u(t)D_{b−}^{α}v(t)|_{a}^{b}.

Further, using the notations of (2.2) we obtain (4.1) from the formula (4.4). Lemma 4.1 implies that the kernel of L_{M} consists of the infinity amount of linear independent functions. Due to the arbitrariness of v from (4.1) we have

ξ_{i}^{−}(g) = ξ_{i}^{+}(g) = 0, i = 1, 2.

Hence f ∈ R(L_{m}). The proof is complete. □

Corollary 4.1. D(L_{m}) is dense in L^{2}(a, b).

Proof. Let g ∈ L^{2}(a, b) be orthogonal to the lineal Dom(L_{m}). Find a function v which is an arbitrary solution of the equation L_{M}v = g. Then for any u ∈ Dom(L_{m}) we have

0 = (u, g) = (u, L_{M}v) = (L_{m}u, v).

By Lemma 4.2 we get v ∈ KerL_{M}. Therefore, g = L_{M}v = 0. The lemma is proved. □

4.1. Proof of Theorem 2.1. By Corollary 4.1 the operator L_{m} is closable in L^{2}(a, b). Then (L_{m})^{∗} = L_{M} in L^{2}(a, b). Hence, it follows that L_{M} is a closed operator. Thereby, H^{2α}(a, b) is the Banach space (see [DS63]). The second part can be proved by checking the axioms of the Hilbert space.

4.2. Proof of Theorem 2.2. Since for any u, v ∈ D(L_{m}), we have

(L_{m}u, v) = (u, L_{m}v),

then by definition [N67] L_{m} is a Hermitian operator. By virtue of Corollary 4.1 the operator L_{m} is a symmetric operator. Thereby, the operator L_{θ} is self–adjoint if

(4.5)  D(L_{θ}) = D(L_{θ}^{∗}).

Finally, the proof of Theorem 2.2 follows from (4.4) by the direct calculations.

5. On some Spectral Properties

Theorem 5.1. For the self–adjoint operator L_{θ} with θ of the form

( \begin{matrix}0 1 0 0 \\ θ_{21} 0 θ_{23} 0 \end{matrix} ) ,

the following facts are true: [(i)] L_{θ}^{−1} is a compact operator in L^{2}(a, b). [(ii)] The spectrum is discrete and real valued, and the system of eigenfunctions is a complete orthogonal basis in L^{2}(a, b).

Proof. (i) If θ_{21} ≠ 0 and θ_{21} ≠ θ_{23} then the inverse operator represents as

L_{θ}^{−1}f(x) = \frac{θ_{21}}{θ_{21} − θ_{23}} \frac{(b − x)^{α}}{(b − a)Γ(α + 1)}I_{a+}^{α+1}f(b) + I_{b−}^{α}I_{a+}^{α}f(x),

and, if θ_{21} = 0 then the representation has the form

L_{θ}^{−1}f(x) = I_{b−}^{α}I_{a+}^{α}f(x), a < x < b.

Indeed, it implies compactness of L_{θ}^{−1} in L^{2}(a, b). (ii) From compactness of the operator L_{θ}^{−1} follows discreteness of the spectrum and that the system of eigenfunctions is a complete orthogonal basis in L^{2}(a, b). Self–adjointness of L_{θ} implies [N67] that all eigenvalues are real. □

Theorem 5.2. Let θ has one of the following forms

(5.1)  ( \begin{matrix}1 0 0 0 \\ 0 1 0 0 \end{matrix} ) , ( \begin{matrix}1 0 0 0 \\ 0 0 0 1 \end{matrix} ) ,

(5.2)  ( \begin{matrix}ρ 1 0 0 \\ 0 0 1 ρ \end{matrix} ) , ( \begin{matrix}0 0 1 0 \\ 0 0 0 1 \end{matrix} ) .

Then for all ρ ∈ ℝ the operator L_{θ} is positive in L^{2}(a, b).

Proof. It is enough to show that

(D_{a+}^{α} [D_{b−}^{α}u] , u) ≥ 0.

Let us calculate

(D_{a+}^{α} [D_{b−}^{α}u] , u) = \frac{1}{Γ(1 − α)} ∫_{a}^{b} ∫_{a}^{x}(x − t)^{−α} \frac{d}{dt}D_{b−}^{α}u(t)dt u(x)dx.

By changing integration order, we obtain

∫_{a}^{b} ∫_{a}^{x}(x − t)^{−α} \frac{d}{dt}D_{b−}^{α}u(t)dt u(x)dx =

= ∫_{a}^{b} \frac{d}{dt}D_{b−}^{α}u(t) ∫_{t}^{b}(x − t)^{−α}u(x)dxdt.

Integrating by parts in the right–side of the last integral, we get

\frac{1}{Γ(1 − α)} ∫_{a}^{b} \frac{d}{dt}D_{b−}^{α}u(t) ∫_{t}^{b}(x − t)^{−α}u(x)dxdt =

= D_{b−}^{α}u(t)I_{b−}^{1−α}u(t)|_{a}^{b}+ (D_{b−}^{α}u, D_{b−}^{α}u) .

Finally, from conditions (5.1) and (5.2) we have

D_{b−}^{α}u(t)I_{b−}^{1−α}u(t)|_{a}^{b} = 0.

Thereby, the theorem is proved. □

Acknowledgements

The first author was supported by the MESRK Grant 0773/GF4 of the Committee of Science, Ministry of Education and Science of the Republic of Kazakhstan. The second author was supported by the MESRK Grant 0819/GF4 of the Committee of Science, Ministry of Education and Science of the Republic of Kazakhstan.

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Niyaz Tokmagambetov Imperial College London 180 Queen’s Gate London, SW7 2AZ United Kingdom n.tokmagambetov@imperial.ac.uk and Al–Farabi Kazakh National University ave. al–Farabi 71 050040 Almaty Kazakhstan niyaz.tokmagambetov@gmail.com Berikbol T. Torebek Institute of Mathematics and Mathematical Modeling Pushkin street 125 050010 Almaty Kazakhstan torebek@math.kz